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Noisefunction
A 'noisefunction' or 'entropy function' is the entropic analogue of the quantum mechanical 'wavefunction'. Theory In Transactional Quantum Mechanics, wave-vectors are replace by wave-tensors which replicate time-evolutions over the spectrum of possible environmental perturbations that could occur. This allows environmental uncertainty to be quantified into the final state evolution, provided that estimates of the environmental noise at relevant frequencies are accurate. Entropy functions are defined on the basis of possible transactions that could have occurred between the system and the environment. The fundamental transaction is defined as an environment-system 'flip-flop' in which the system has one or more of its eigen-components undergo a magnitude-flip either subtracting or adding one quantum of thermal energy from/to the system. Structure A noisefunction of a thermal environment is structured in a way that correlates to the energy wavefunction of a quantum system. Wave Vectors The states of quantum systems are represented by wave-vectors, which delineate the magnitude and phase of the wave components of the superposition state that correspond to each eigenstate (when defined in the eigenbasis). A system with N possible eigenstates will then have a wave-vector of dimension N. Example: a 4-dimensional quantum system in an equal superposition of all four eigenstates still experiences time evolution between all of its relative amplitudes, because although eigenstates do not evolve in magnitude, they all evolve in phase at their own frequencies, hence after a time duration of one period for the base frequency 'ω_0', the other frequencies will have completed more than one revolution in phase and will for example end up as below: where here ω_3 has completed 2.25 oscillations in the time it took ω_0 to complete one, and ω_1 and ω_2 have respectively completed 1.333... and 1.75 oscillations. Transition Matrices Quantum operators act on superposition states in ways that can be represented as matrix multiplications performed on the wave vector. Transition matrices quantify the relative changes that a quantum operator makes to each eigenstate. For example, a T_01 matrix might flip the magnitudes of the ω_0 and ω_1 eigenstates: While a T_0213 matrix might flip the ω_0 and ω_2 amplitudes as well as the ω_1 and ω_3 amplitudes: Note here that the real components of the matrix change the real components of the amplitudes (the magnitude), while imaginary components are responsible for rotations of the phase. For example a quantum operation that rotates the phase of all components by 180° could be represented by R_ππππ: (the sign of a wave-vector amplitude is irrelevant to its magnitude, since magnitudes are calculated by the modulus of the amplitude) Noise Tensor A noise tensor is a multi-dimensional 'vector analogue' that encodes entropically probable transitions into a set of possible transition matrices that could occur (or could have occured) during the (pre-)evolution of a state. These transitions, when applied to the wavefunction, alter the state trajectory in accordance with the environment perturbations encoded in each matrix. Applying the full tensor to a wave-function generations a set of possible wave-functions, which can then be summed over to approximate an 'average wavefunction' given the environmental uncertainties. Advantage If environmental uncertainties are extreme, then the noise function is not particularly advantageous. The sum of many possible pertubations will become a sum over a near-random spread of state paths, which average towards zero-information - meaning random chance. This actually represents an accurate prediction of the state, since in such scenarios the pure-state assumptions of traditional quantum mechanics break down and a maximally mixed-state of 50% purity is the correct state. However, the strength of this approach is more relevant for the case when environmental fluctuations are rare but still fully projective. In this case, as long as reasonable estimates of the noise spectral density of the environment can be made, then an accurate noise tensor can be calculated and state evolution can essentially quantify the decoherence in a systematic way.Category:Entropy Category:Quantum Mechanics